Convergence in probability and expectation

Question

Let $\mathbb EX$ exist and $X_n$ converges in probability to $X$. How to prove that $\mathbb EX_n → \mathbb EX$ then and only when $\mathbb E| X_n - X | → 0$?

Answer 1

This is not true. Consider $[-1,1]$ with the measure $P(A)=\frac {\lambda(A)} 2$ where $\lambda$ is Lebesgue measure. Let $X_n(x)=n^{2} xI_{(-\frac 1 n, \frac 1 n)}(x)$, and $X\equiv0$. Then $EX_n =EX=0$ for all $n$ and $X_n \to X$ almost surely, hence in probability. But $E|X_n|=1$ for all $n$.